I opened my filter (tetra quiet flow 10) and saw that the carbon filter is pretty dirty. I also saw strange string-like things trailing from the cartridge. Is this normal? Should I be cleaning the cartridge more often?

Hmmmmm... could be most likely algea, dont change the filter cartridge, just see if you can take some of the stuff off with your hand every once in a while. I dont think it would be a problem as long as your water isnt cloudy. I think if it was algea your main concern would be the algea dying and then polluting the water cuz its inside your filter. So keep an eye on it and manually try to keep it under control and see what happens. Not 100% positive about it, so we will see what other people say...... good luck!!